DFT insights into the electronic structure, mechanical behaviour, lattice dynamics and defect processes in the first Sc-based MAX phase Sc2SnC

Here we employed the density functional theory calculations to investigate some physical properties of first Sc-based MAX phase Sc2SnC including defect processes to compare with those of existing M2SnC phases. The calculated structural properties are in good agreement with the experimental values. The new phase Sc2SnC is structurally, mechanically and dynamically stable. Sc2SnC is metallic with a mixture of covalent and ionic character. The covalency of Sc2SnC including M2SnC is mostly controlled by the effective valence. Sc2SnC in M2SnC family ranks second in the scale of deformability and softness. The elastic anisotropy level in Sc2SnC is moderate compared to the other M2SnC phases. The hardness and melting point of Sc2SnC, including M2SnC, follows the trend of bulk modulus. Like other members of the M2SnC family, Sc2SnC has the potential to be etched into 2D MXenes and has the potential to be a thermal barrier coating material.


Computational methods
The DFT calculations were executed with the CASTEP code 18 . The Perdew-Burke-Ernzerhof (PBE) functional in the frame of generalized gradient approximation (GGA) was employed to estimate the electronic exchange-correlation energy 19 . Ultra-soft pseudo-potential developed by Vanderbilt was used to model the interactions between electrons and ion cores 20 . The Monkhorst-Pack (MP) scheme with a Γ-centered k-point mesh of 15 × 15 × 3 grid is employed to integrate over the first Brillouin zone in the reciprocal space of hexagonal unit cell of Sc 2 SnC 21 . A cutoff energy of 700 eV was chosen to expand the eigenfunctions of the valence and nearly valence electrons using a plane-wave basis. During the geometry optimization, both the total energy and internal forces were minimized with the Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm 22 . To achieve the self-consistent convergence the difference in the total energy is kept less than 5 × 10 -6 eV/atom, the maximum ionic Hellmann-Feynman force less than 0.01 eV/Å, maximum ionic displacement less than 5 × 10 -4 Å, and maximum stress less than 0.02 GPa.
The elastic properties are investigated using finite-strain theory as embedded in the CASTEP code 23 . In this method, a predetermined value for strain is used to relax all the free parameters and compute the stress. For elastic calculations, the convergence criteria are set as: the difference in total energy less than 10 -6 eV/atom, the maximum ionic Hellmann-Feynman force less than 2 × 10 -3 eV/Å, and the maximum ionic displacement less than 10 -4 Å. The finite-strain theory as implemented in CASTEP has been successfully employed to calculate the elastic properties of numerous systems [24][25][26][27][28][29][30][31][32][33][34][35] .
Lattice dynamic properties such as phonon dispersion and phonon density of states are calculated using a 3 × 3 × 1 supercell defined by cutoff radius of 5.0 Å employing the finite displacement supercell method within the code. A 35 × 35 × 7 k-point mesh was used to calculate the electronic charge density distribution and the Fermi surface. Defect calculations were carried out with a 3 × 3 × 1 supercell of 72-atomic site (36 M, 18A, and 18 C) under constant pressure. To determine the potential interstitial sites a thorough computational search was performed examining all potential interstitial sites. The calculations with large supercell require comparatively small cutoff energy and moderate k-point mesh. A cutoff energy of 350 eV and a k-point mesh of 3 × 3 × 1 grid in the MP scheme are used for the supercell defect calculations.

Results and discussions
Structural properties. Commonly with other MAX phases Sc 2 SnC crystallizes in the hexagonal space group P6 3 /mmc (No. 194) 12 . Each unit cell of Sc 2 SnC contains two formula units and eight atoms (refer to Fig. 1a). The Sc atoms occupy the 4f Wyckoff site with the fractional coordinates (1/3, 2/3, z), the Sn resides in the 2d atomic site with the fractional coordinates (1/3, 2/3, 3/4) and the C atoms are accommodated at the 2a Wyckoff position with the fractional coordinates (0, 0, 0). The atomic sites of Sc 2 SnC with theses fractional coordinates are also valid for all 211 MAX phases. The optimized lattice parameters are listed in Table S1 in the supplementary section along with those of all M 2 SnC MAX phases including experimental values 10,12,16,17,36 . The predicted values for Sc 2 SnC are very good agreement with the experimental values, supporting the validity of the present investigation. In previous studies 16, 17 , we observed that the lattice constants of Sn-based 211 MAX phases maintain a good relation with the crystal radius of M-atoms. Both lattice constants a and c increase almost linearly with the crystal radius of M-atoms. The Sn-based new compound Sc 2 SnC also maintains this relationship (see Fig. 1b and c). Band structure and DOS. The electronic band structure of Sc 2 SnC was calculated along the high symmetry directions in the first Brillouin zone (refer to Fig. 2a). It reveals the metallic characteristics of Sc 2 4 . A notable feature in the band structure is the considerable anisotropic nature with low energy dispersion along the c-axis. This is apparent from the reduced dispersion along the short H-K and M-L directions. The anisotropic nature of the band structure near and below the Fermi level indicates that the electrical conductivity is as anisotropic for the Sc 2 SnC as for the other M 2 SnC MAX phases.
To obtain more information on the chemical bonding in Sc 2 SnC, the total and partial densities of states were calculated (refer to Fig. 2b). In this figure, the broken vertical green line refers to the Fermi level E F , which is located to the left of a pseudogap in the total DOS. It is one of the indications of the structural stability of Sc 2 SnC. The proximity of E F to the pseudogap can lead to more structural stability of the compounds of mixed bonding character 37,38 . Comparing the position of E F relative to the pseudogap for all existing M 2 SnC MAX phases the structural stability should follow the order: Nb 2 SnC > Ti 2 SnC > Lu 2 SnC > Zr 2 SnC > Hf 2 SnC > Sc 2 SnC > V 2 SnC. The main contribution to the total DOS at E F comes from the d-orbital of Sc. The d-resonance at the surroundings of E F and the finite value of the total DOS at E F indicates the metallic character of Sc 2 SnC and this is a common feature of MAX phases. The total DOS of Sc 2 SnC at E F is 3.10 states/eV-uc, which is about half of V 2 SnC (6.12 states/eV-uc) and between the range (2.35-3.93 states/eV-uc) of other M 2 SnC phases 4,17 . Above the E F , the antibonding states arise due to d-orbitals of M atom in Sc 2 SnC in similar to other M 2 SnC MAX phases.
The valence band of Sc 2 SnC is divided into two main parts. The lower part is situated between −10.4 eV and −4.9 eV, which contains a distinct peak and a flat region. The peak originates as a result of hybridization between Sc 3d and C 2s orbitals, indicating covalent Sc-C bonding. The flat region arises due to the main contribution of Sn 5s electrons. The upper valence band consists of two distinct peaks. The highest peak close to E F is also due to the hybridization of Sc 3d electrons with C 2p electrons. Hybridization between Sc 3p and Sn 5p near the E F also contributes to the highest peak of the total DOS. This hybridization leads to the formation of the Sc-Sn covalent bond between Sc and Sn. This bond is not as strong as Sc-C because the corresponding peak is closer to the Fermi level. The lowest peak centered at −3.3 eV arises due to the interaction between Sc 3d and C 2p states.   Charge density. The contour map of the electron charge distribution among the constituent atoms in a compound is a way to understand the nature of atomic bonding in the material. The contour map for Sc 2 SnC is given in Fig. 2c. It can be observed that the charge distributions around the atoms have created an almost spherical electron cloud and its intensity determines the amount of charge accumulation. The amount of charge accumulated around the Sc atom is 0.53e, while the amount of charge accumulated around the M atom of other M 2 SnC phases is between 0.28 and 0.45e 4,17 . Clearly, the maximum charge accumulates around the Sc atom among all M-atoms in M 2 SnC phases. The minimum charge accumulates around Hf 4 . The electron cloud of Sc-charge overlaps with that of C-charge and slightly edges with that of Sn-charge, which indicates the stronger covalent Sc-C and weaker covalent Sc-Sn bonding, respectively. The spherical distribution of charge around the atoms is an indication of some ionic character in chemical bonds in Sc 2 SnC. The contour map of electron charge distribution for Sc 2 SnC is almost identical to those of other M 2 SnC phases. Mulliken population. Population analysis in CASTEP is carried out using a projection of the planewave (PW) states onto a linear combination of atomic orbitals (LCAO) localized basis using a method developed by Sanchez-Portal et al. 40 . Population analysis of the resultant projected states is then accomplished using the Mulliken formalism 41 . This analysis provides the Mulliken charge, bond population and bond length in a bulk material. Mulliken charge associated with a given atom, A, can be determined as: where P μν (k) is the density matrix and S μν (k) is overlap matrix. The bond population between two atoms A and B can be calculated as: The Mulliken charge measures the effective valence from the absolute difference between the formal ionic charge and the Mulliken charge on the atomic species. Table S2 lists the effective valence, bond population, and bond length between different atoms in Sc 2 SnC and existing M 2 SnC MAX phases. The pure valence states for transition metals Sc, Ti, V, Zr, Nb, Lu, and Hf in M 2 SnC MAX phases are 3d 1 , 3d 2 , 3d 3 , 4d 2 , 4d 4 5s 1 , 5d 1 , 5d 2 , respectively. It is observed that the effective valence largely depends on the d-orbital electrons of the transition metals. It increases when the transition metal moves from the left to the right in the periodic table. Its non-zero positive value is an indication of mixed covalent and ionic nature in chemical bonds. Its progression towards zero value indicates an increase in the level of ionicity. Its zero value implies an ideal ionic character in a chemical bond. Its progression from zero with a positive value indicates an increase in covalency level of chemical bonds. Based on the effective valence the covalency of M 2 SnC increase when M atoms move from the left to the right in the periodic table.
Bond population is another indication of bond covalency in a crystal as a high value of bond population in essence indicates a high degree of covalency in the chemical bond. The M-C bond in the MAX phases is mainly covalent bond. The bond population of M-C bond in each M 2 SnC MAX phases is positive except in Lu 2 SnC. The bond population of M-C bond in M 2 SnC deceases when the M atom moves from the left to the right in the periodic table, indicating the decrease in covalency. Actually, effective valence and positive bond population collectively control the covalency of crystalline solids. The bulk modulus is mostly controlled by the bond covalency. Between effective valence and positive bond population, which is most influential in bond covalency? This can be verified with the bulk modulus. It is observed from the Fig. 4a in the next section that the bulk modulus changes according to the effective valence. Therefore, it can be concluded that the effective valence mainly controls the covalency level in the studied compounds. The bond length of covalent M-C bond deceases when the M atom moves from the left to the right in the periodic table. It is clear that the shorter the covalent bond length, the Single crystal elastic constants. Elastic constants are the fundamental tools for accessing the mechanical behavior of crystalline solids. MAX phases have five independent elastic constants C ij due to their hexagonal crystal symmetry. These are C 11 , C 33 , C 44 , C 12 and C 13 . In addition they have one more dependent elastic constant C 66 , which depends on C 11 and C 12 and C 66 = (C 11 -C 12 )/2. First of all, the elastic constants justify the mechanical stability of compounds obeying Born criteria. For hexagonal systems these criteria are as follows 42 : The calculated elastic constants of Sc 2 SnC are listed in Table S3 and shown in Fig. 3a along with CASTEPderived elastic constants for existing M 2 SnC phases for comparison. Sc 2 SnC meets the above conditions by its elastic constants like its predecessors M 2 SnC and ensures its own mechanical stability like its predecessors.
The elastic constants C 11 and C 33 represent the resistance to linear compression, whereas other constants such as C 12 , C 13 , and C 44 represent the resistance to shape change. Indeed, C 11 and C 33 represent the stiffness along the crystallographic a-and c-axis, respectively. The stiffness of Sc 2 SnC is slightly larger along the a-axis than along the c-axis, which is also observed for Ti 2 SnC, Nb 2 SnC and Hf 2 SnC. For the remaining M 2 SnC phases, V 2 SnC, Zr 2 SnC, and Lu 2 SnC, the stiffness along the c-axis is slightly larger than that in the a-axis. The difference between C 11 and C 33 quantifies the level of elastic anisotropy in crystals relating to the crystallographic axis. Accordingly, V 2 SnC, Nb 2 SnC, and Hf 2 SnC are elastically more anisotropic than Sc 2 SnC, Ti 2 SnC, Zr 2 SnC, and Lu 2 SnC. The new phase Sc 2 SnC ranks fourth in the view of both less and high anisotropy in the M 2 SnC family of seven members.
Shear elastic constants C 12 and C 13 reciprocally lead to a functional stress component along the crystallographic a-axis with a uniaxial strain along the crystallographic b-and c-axis, respectively. This stress component takes the measurements of the shear deformation resistance of a compound along the crystallographic b-and c-axis, when stress is applied along the a-axis. The Nb 2 SnC phase is most capable of resisting such deformation, whereas Lu 2 SnC will easily deform under an equal stress along the a-axis. The new compound Sc 2 SnC will be the second in rank in M 2 SnC systems that will be easily deformed if a rank of deformation resistance of M 2 SnC system is made: Nb 2 SnC > V 2 SnC > Hf 2 SnC > Ti 2 SnC > Zr 2 SnC > Sc 2 SnC > Lu 2 SnC.
The elastic constant C 44 provides an indirect measure of the indentation hardness of a material. A low value of C 44 indicates higher shearability and low hardness of a compound. High shearability and low hardness are related to better machinability of a compound. Due to low value of C 44 , Lu 2 SnC has highest shearability among the seven M 2 SnC MAX phases. The new material Sc 2 SnC should be the second in rank in the M 2 SnC systems for shearability.
The symmetry condition C 66 = (C 11 -C 12 )/2 represents an important consequence in hexagonal crystals. C 66 serves as the shear constant on the (100) plane in a [010] direction, while (C 11 -C 12 )/2 stands for the shear constant on the (110) plane in a [110] direction. Therefore, the elastic shear constant is the same for all planes in the [001] zone, independent on the specific shear plane or shear direction, which is known as the transverse isotropy. This means that the elastic constants are invariant for arbitrary rotation around the z-axis: in the xy plane, the hexagonal crystals are elastically isotropic, which we will observe in a subsequent section.
Bulk elastic moduli. Elastic moduli are the most important elastic parameters that assess the mechanical behavior of crystalline solids. Calculated elastic moduli are listed in Table S3 and shown in Fig. 3b. Bulk modulus B and shear modulus G can be derived from the elastic constants C ij using Voight-Reuss-Hill approximations [43][44][45] . A detailed discussion of these methods for hexagonal crystals is found in a recent study 46 . The bulk modulus of (3) C 11 , C 33 , C 44 > 0; C 11 > |C 12 | and (C 11 + C 12 )C 33 > 2C 13 C 13 www.nature.com/scientificreports/ a crystal depends microscopically on the nature of its bond such as length and type. In the case of the studied compound it is observed that it is also controlled by the total effective valence of the crystal (see Fig. 4a). Bulk modulus is a measure of the resistance to uniform compression of a material and it is linked to chemical composition and crystal structure. Among M 2 SnC phases, the new phase Sc 2 SnC possesses the second lowest value for B. The highest value is assigned to Nb 2 SnC and the lowest value is associated with Lu 2 SnC. Therefore, Sc 2 SnC will be compressed more easily as compared to existing M 2 SnC phases except Lu 2 SnC. The shear modulus G is concerned with the deformation of a solid material when it experiences a force parallel to one of its surfaces while its opposite face experiences an opposing force such as friction. G maintains a good relationship with C 44 .
Here, it is reflected and Sc 2 SnC secure the second rank in the scale of lowest value of G similar to C 44 . B and G collectively prescribe an important parameter known as Pugh's ratio (defined as B/G) that evaluates a necessary mechanical behavior of crystalline solids 47 . Generally, a material is either considered brittle or ductile. The brittle materials have a value less than 1.75 and ductile materials possess a value greater than 1.75. Materials with a B/G value above or below this borderline value behave in a less ductile or brittle manner. Accordingly, Sc 2 SnC is a brittle material similar to Ti 2 SnC, Zr 2 SnC, Lu 2 SnC and Hf 2 SnC while V 2 SnC and Nb 2 SnC exhibit ductility (refer to Fig. 4b).
Poisson's ratio v is another important parameter and can be derived from B and G: v = (3B-2G)/(6B + 2G). Similar to Pugh's ratio, Poisson's ratio can serve as a predictor for distinguishing brittle from ductile materials. Poisson's ratio v with a value less than 0.26 identifies the materials as brittle ones and with a higher value the ductile ones 48 . The Poisson's ratio has classified the M 2 SnC MAX phases into brittle and ductile groups, consistently with Pugh's ratio above. That is, the Sc 2 SnC can be considered brittle. The Poisson's ratio can also identify the interatomic forces between atoms in a solid 49 . When the Poisson's ratio of a solid is between 0.25 and 0.50, the interatomic forces between the atoms of that solid will be the central forces and if the Poisson's ratio is outside of this range the interatomic forces will be the non-central forces 50 . A central force is a force (possibly negative) that points directly from a particle to a certain point in space, the center and whose magnitude depend only on the distance of the particle from the center while a non-centrifugal force is a force between two particles that is not directed along their connecting line.    Elastic moduli B and G also provide another essential property, the Young's modulus E via the relation, E = 9BG/(3B + G). The Young's modulus of a material is a useful property for predicting the behaviour of the material when subjected to a tensile force. Stiffness of a material mostly depends on its Young's modulus. Higher Young's modulus is an indication of higher stiffness. In the family of M 2 SnC MAX phases, Ti 2 SnC is the stiffest material and Lu 2 SnC is the softest one. The newly synthesized Sc 2 SnC ranks second on the scale of softness: Lu 2 SnC > Sc 2 SnC > Nb 2 SnC > Zr 2 SnC > V 2 SnC > Hf 2 SnC > Ti 2 SnC. The Young's modulus of MAX phases can be related to the exfoliation energy. The smaller the Young's modulus, the softer the system and hence the lower the exfoliation energy and the higher the possibility of etching into 2D MXenes 52 . The four MAX phases Ti 2 AlC, Ti 2 AlN, V 2 AlC, and Nb 2 AlC in the 211 family are exfoliated experimentally into MXenes 53 . Their theoretical Young's moduli 54 range from 262 to 312 GPa and exfoliation energies 53 range from 0.164 to 0.205 eV/Å 2 . V 2 AlC has the highest Young's modulus (~ 312 GPa) and consequently has the highest exfoliation energy (0.205 eV/Å 2 ). As the Young's moduli of the Sn-based 211 MAX phases under study range from 152 to 219 GPa, their exfoliation energies can be expected to be lower than 0.205 eV/Å 2 . Very recently, the exfoliation energies of Sc 2 SnC, Ti 2 SnC, V 2 SnC, Zr 2 SnC, Nb 2 SnC, and Hf 2 SnC are calculated to be 0.131, 0.164, 0.137, 0.157, 0.150, and 0.158 eV/Å 2 , respectively 55 . These values lie within the range between 0.131 and 0.164 eV/Å 2 , which are lower than the range of 0.164 and 0.205 eV/Å 2 . As the Young's modulus of Lu 2 SnC is lowest in the M 2 SnC phases considered here, its exfoliation energy can be expected to lie within this range. The lower the exfoliation energy, the higher the possibility to be etched experimentally into 2D MXenes. Therefore, Lu 2 SnC and other M 2 SnC phases considered here are more likely to be etched into 2D MXenes than V 2 AlC. Further, the Young's modulus E has a good relation to the thermal shock resistance R: R ∝ 1/E 56 . The lower the Young's modulus, the better the thermal shock resistance. A material of higher thermal shock resistance (i.e., lower Young's modulus) has the potential to be used as a TBC material. The Young's modulus of Sc 2 SnC and other M 2 SnC MAX phases are lower than that of a potential TBC material TiO 2 whose Young modulus is 283 GPa 57 . Therefore, Sc 2 SnC and other existing M 2 SnC phases have possibility to be TBC materials if they also have high thermal expansion coefficient and melting point, low thermal conductivity, and good oxidation resistance.
Elastic anisotropy. The study of elastic anisotropy is important as it influences a variety of physical processes including the development of plastic deformation in crystals, microscale cracking in ceramics, and plastic relaxation in thin-film metallics 58 . For hexagonal crystals like MAX phases the shear anisotropy factors A i (i = 1, 2, 3) are studied extensively 16,27,28,31,59 . The equation that determines the shear anisotropy factor A 1 , for the {100} shear planes between the < 011 > and < 010 > directions, is A 1 = (C 11 + C 12 + 2C 33 4C 12 )/6C 44 ; the equation of A 2 , for the {010} shear planes between < 101 > and < 001 > directions, is A 2 = 2C 44 /(C 11 C 12 ); and the equation of A 3 , for the {001} shear planes between < 110 > and < 010 > directions, is A 3 = (C 11 + C 12 + 2C 33 4C 13 )/3(C 11 C 12 ). Deviation of A i from unity ΔA i (= A i ~ 1) quantifies the degree of shear anisotropy of crystals. The calculated A i is listed in Table S4 and the anisotropy level ΔA i is shown in Fig. 4c. Considering the average on all the planes, Ti 2 SnC is elastically less anisotropic and Nb 2 SnC is elastically highly anisotropic. Sc 2 SnC ranks third in view of less anisotropy in the M 2 SnC family: Nb 2 SnC > Hf 2 SnC > Zr 2 SnC > Lu 2 SnC > Sc 2 SnC > V 2 SnC > Ti 2 SnC. Individually, in the {100} shear planes Nb 2 SnC is highly anisotropic; in the {010} shear planes Nb 2 SnC is again highly anisotropic and in the {001} shear planes Hf 2 SnC is highly anisotropic.
The anisotropy level in the hexagonal crystals like MAX phases can also be quantified by another anisotropy factor named compressibility anisotropy factor and it is defined as k c /k a = (C 11 + C 12 2C 13 )/(C 33 C 13 ) 48 . Here, k a and k c are the linear compressibility coefficients along the a-and c-axis, respectively. Deviation of k c /k a from the unity Δ(k c /k a ) (= k c /k a ~ 1) quantifies the degree of the compressibility anisotropy of crystals. The calculated k c /k a is listed in Table S4 and Δ(k c /k a ) is shown in Fig. 4c. The compressibility anisotropy level is highest in V 2 SnC and lowest in Nb 2 SnC. Sc 2 SnC ranks in the middle in the M 2 SnC family of seven members. If k c /k a > 1, the material is more compressible along the c-axis than along the a-axis. Therefore, Sc 2 SnC, Ti 2 SnC and Lu 2 SnC are slightly more compressible along the c-axis than along the a-axis while V 2 SnC, Zr 2 SnC, Nb 2 SnC and Hf 2 SnC are compressed more easily along the a-axis than along the c-axis.
There are some anisotropy factors such as percentage anisotropy factors A B% and A G% based on the bulk and shear moduli within the Voigt and Reuss limits, which are applicable for all types of crystals. A B% measures anisotropy in compression while A G% measures anisotropy in shear. These two factors are defined as 51 . The calculated values are listed in Table S4. Both these factors assign zero value for isotropic crystals and their positive values indicate the anisotropy level in crystals. A B% is highest for V 2 SnC and lowest for Ti 2 SnC while A G% is highest for Hf 2 SnC and lowest for Ti 2 SnC. Sc 2 SnC ranks fourth on the A B% scale and second on the A G% scale in terms of minimum anisotropy. Universal anisotropy factor A U is also applicable for all types of crystals. It is defined as A U = 5(G V /G R ) + (B V /B R )6 ≥ 0 17 . Its zero value corresponds to isotropic crystals and a positive value implies the anisotropy level in crystals. The calculated values are listed in Table S4. Hf 2 SnC has the highest value of A U and Ti 2 SnC possesses the lowest value. Sc 2 SnC has the second lowest value of A U .
The 2D and 3D graphical representations of the directional elastic properties of materials are visualization of elastic anisotropy in crystals. ELATE is an open-source software 60 , which allows the direct visualization of anisotropy level in Young's modulus (E), linear compressibility (β), shear modulus (G) and Poisson's ratio (v) on the 3D spherical plot, as well as 2D projections on the (xy), (xz) and (yz) planes. Uniform circular 2D and spherical 3D graphical representations are the indications of isotropic nature of crystals. As the MAX phases are hexagonal crystals, they are elastically isotropic in the xy plane. It is evident that the 2D presentation of E, β, G and v of Sc 2 SnC in the xy plane in Fig. 5 Fig. 5 is indicating the elastic anisotropy of Sc 2 SnC in those planes. The greater the deviation from the round shape, the higher the anisotropy level in the crystals in that plane. In 2D and 3D presentations, ELATE uses maximum two colors for E and β and maximum three colors for G and v. The E and β are functions of a single unit vector a(θ, ϕ) while G and v depend on two orthogonal unit vectors a(θ, ϕ) and b(θ, ϕ, χ) (a in the direction of the stress applied while b in the direction of measurement). The spherical coordinates θ, ϕ, and χ can be defined as 0 ⩽ θ ⩽ π, 0 ⩽ ϕ ⩽ 2π, and 0 ⩽ χ ⩽ 2π. Therefore, E, β, G and v can be expressed as E (θ, ϕ), β(θ, ϕ), G(θ, ϕ, χ) and ν(θ, ϕ, χ). G(θ, ϕ, χ) and ν(θ, ϕ, χ) are represented in 3D space via plotting two surfaces f and g each with the spherical (θ, ϕ) coordinates. The surfaces f and g represent the minimal and maximal values over all possible values of χ: f (θ, ϕ) = min χ X(θ, ϕ, χ) and g(θ, ϕ) = max χ X(θ, ϕ, χ), respectively. The surface g encloses the surface f. For this reason, g is plotted in translucent blue color in www.nature.com/scientificreports/ for elastic properties, which are not required to be along the crystallographic axes of the material. Additionally, it reports a measurement of the anisotropy A X of each elastic modulus X, which is defined below: The results are listed in Table 1. It is evident that Young's modulus exhibits maximum anisotropy for Nb 2 SnC and minimum for Ti 2 SnC and Sc 2 SnC ranks second in view of minimum anisotropy. Anisotropy in linear compressibility is maximum for V 2 SnC and minimum for Ti 2 SnC; Sc 2 SnC ranks third in scale of minimum anisotropy. Anisotropy in shear modulus is highest for Hf 2 SnC and lowest for Ti 2 SnC; Sc 2 SnC ranks second in view of minimum anisotropy. Maximum anisotropy of Poisson's ratio is observed in Hf 2 SnC and minimum in Ti 2 SnC, whereas Sc 2 SnC ranks third in view of minimum anisotropy. The lowest anisotropy is observed for Ti 2 SnC in the M 2 SnC family considering all indicators.
Theoretical hardness. Hardness is the property of a material that facilitates it to resist plastic deformation, penetration, indentation and scratching. Therefore, hardness is important from an engineering point of view because the resistance to wear by either abrasion or corrosion by steam, oil and water usually increases with hardness. Theoretical modeling for hardness calculation of partially metallic compounds like ternary MAX phases is difficult. Gou  where P µ denotes to the positive Mulliken bond overlap population of the μ-type bond, P µ ′ represents the metallic population that is derived from the unit cell volume V and the number of free electrons in the cell, n free with the formula, P µ ′ = n free V , here n free = E F E P N(E)dE and E P and E F define the energies at the pseudogap and at the Fermi level, respectively, and v µ b is the bond volume of a μ-type bond calculated using the equation When a compound has a positive bond population for multiple bonds, the following equation is used to calculate its Vickers hardness: where n µ represents the number of μ-type bonds. Table S5 lists the Vickers hardness of Sc 2 SnC and other M 2 SnC MAX phases. Sc 2 SnC is harder than Lu 2 SnC and Zr 2 SnC and softer than Ti 2 SnC, V 2 SnC, Nb 2 SnC and Hf 2 SnC. We have found two sets of experimental Vickers hardness for Ti 2 SnC, Zr 2 SnC, Hf 2 SnC, and Nb 2 SnC 36,62 . Experimental values show deviations from one set to another except for Ti 2 SnC. This can be due to sample purity and errors induced by the instruments. Furthermore, the experiment is performed with a sample that contains In general, the hardness of a compound has a better relationship with its shear and Young's modulus than bulk modulus 3 . We have plotted the Vickers hardness of M 2 SnC MAX phases in Fig. 4d along with their elastic moduli. For the M 2 SnC MAX phases, it is observed that the hardness follows the trend of bulk modulus instead of shear and Young's modulus. Further verification is needed to determine whether this trend continues for the carbide MAX phase with a specific A-group element.
Lattice dynamics. The subject of lattice dynamics is the study of the vibrations of the atoms in a crystal. The vibrations of the atoms are related to many important physical properties such as lattice thermal conductivity, minimum thermal conductivity, Debye temperature, melting point, phonon dispersion, phonon DOS etc. These properties are investigated for newly synthesized Sc 2 SnC to compare with existing M 2 SnC phases.
Debye temperature. Debye temperature is a characteristic temperature at which the highest-frequency mode (and hence every possible mode) is excited. It is related to many physical properties such as thermal expansion, thermal conductivity, specific heat and lattice enthalpy. The Anderson method is simple and rigorous way to calculate the Debye temperature of crystalline materials using the equation 63 : All symbols bear the conventional meanings and v m refers to the average sound velocity, which can be determined using the following equation: Here, v l and v t are the longitudinal and transverse sound velocities, respectively. They can be calculated from the bulk and shear moduli B and G using the equations: The calculated values of θ D for M 2 SnC MAX phases are listed in Table S6 along with the relevant quantities. The Debye temperature of Sc 2 SnC is the third highest in the M 2 SnC family. In this family, Ti 2 SnC possesses the highest Debye temperature and Lu 2 SnC has the lowest Debye temperature. The Debye temperature of M 2 SnC phases largely depends on the sound velocities and follows the trend of change of sound velocities with the transition metal M (refer to Fig. 6a).
Melting point. Melting point of hexagonal crystals like MAX phases can be calculated from elastic constants using: T m = 354 + 1.5(2C 11 + C 33 ) 64 . The calculated values are listed in Table S6. The new phase Sc 2 SnC possesses the second lowest melting point. The highest melting point is obtained for Ti 2 SnC and the lowest melting point is observed for Lu 2 SnC. A higher melting point indicates greater interatomic forces in crystals. Interatomic forces mainly control the bulk elastic properties of crystalline solids. Thus, a relationship can exist between the elastic modulus and the melting temperature of the crystals. Considering this we plotted the elastic moduli and the  Fig. 6b. It is observed that the melting point has better correlation with bulk modulus B and Young's modulus E than shear modulus G.
Lattice thermal conductivity. The lattice thermal conductivity arises from contributions of phonons of all frequencies. The knowledge of lattice thermal conductivity is important to determine the applicability of a material for use in high temperature environments. Recently, room temperature lattice thermal conductivity of two MAX phases Zr 2 SeC and Zr 2 SC are reported, which are 75 and 80.7% of the total thermal conductivity of the compounds, respectively 11 . So, the lattice thermal conductivity of metallic compounds like MAX phases provides the concept of total thermal conductivity in a computationally tractable way. For this reason, many authors have reported lattice thermal conductivity of many compounds including MAX phases 11,[65][66][67][68] Encouraged by these reports we have calculated the lattice thermal conductivity of the newly synthesized Sc 2 SnC MAX phase. Here, the Slack model is used to calculate the lattice thermal conductivity of Sc 2 SnC as MAX phases have dual characteristics of metals and ceramics 69 . The following equation is used in this model: The details of this model are given in a recent study 70 . The room temperature lattice thermal conductivity of Sc 2 SnC and other M 2 SnC phases are listed in Table S6 and their temperature dependency is shown in Fig. 7a. The Debye temperature of the M 2 SnC phases is calculated to range from 300 to 525 K. This implies that all vibrational modes will be active above these temperatures for the M 2 SnC MAX phases. Consequently, from these temperatures and above the phonon contribution becomes dominant in the total thermal conductivity. Here, we report the lattice thermal conductivity of M 2 SnC MAX phases in the temperature range of 300 to 1100 K. In this temperature range, the electronic contribution to the total thermal conductivity should be insignificant. The room temperature (300 K) thermal conductivity of 60 W K -1 m -1 , measured in Ti 2 SC is the highest to date, despite the fact that its electrical conductivity (1.926 × 10 6 Ω -1 m -1 ) is relatively poor. This is due to the large contribution from phonons 71 . For the temperatures at and above 300 K, the lattice contribution should dominate the total thermal conductivity of MAX phases. For comparison, we have the literature values for lattice thermal conductivity of Nb 2 SnC 65 , which are also plotted in Fig. 7. The present values show good agreement with the literature values. It is observed that Sc 2 SnC has the third highest lattice thermal conductivity in the entire temperature range and Ti 2 SnC and Nb 2 SnC possess the highest and lowest values, respectively. The lattice thermal conductivity of M 2 SnC MAX phases decrease gradually with the increase of temperature. The rate of decrease is almost similar for all M 2 SnC phase. Sc 2 SnC should be suitable candidate as a TBC material as other M 2 SnC phases are 17 .
Minimum thermal conductivity. Thermal conductivity decreases with increasing temperature. Thus, the minimum value of thermal conductivity is significant for the application of materials at high temperature conditions, for instance, materials selection and design for thermoelectric, thermal barrier coating and other thermal management applications. The concept of a minimum thermal conductivity, κ min , carried by the atomic vibrations of any solid material led to the development of different models. The Clarke model has become popular for determining the minimum thermal conductivity of solids via the Eq. 72 :  7). The calculated value of κ min is listed in Table S6. The phase Sc 2 SnC has the third highest value for κ min . The highest value is found for V 2 SnC and the lowest value is observed for Lu 2 SnC. The ultralow minimum thermal conductivity of 1.25 W/m-K is used for screening the appropriate materials for TBC application 73 . The values of κ min for M 2 SnC MAX phases are lower than this optimum value. Therefore, all M 2 SnC phases should be promising TBC materials with greater possibility for Lu 2 SnC. The minimum thermal conductivity has a linear correlation with the Debye temperature for M 2 SnC phases (Fig. 7b).

Phonon dispersion and phonon DOS.
It is important to study the phonon dispersion and phonon density of states (DOS) to verify the dynamical stability of the crystalline solids. The calculated phonon dispersion is shown in Fig. 8a. For a dynamically stable crystal, there are always three phonons with zero frequency at Γ-point, which corresponds to k = 0 in reciprocal space. The phonon branches starting at ω(k) = 0 are called acoustic phonon dispersion curves 74 . In the case of Sc 2 SnC (refer to Fig. 8a) the acoustic branches start at ω(k) = 0 and consequently indicate the dynamical stability of Sc 2 SnC. The phonons, whose frequencies are non-zero at the Γ point, are called optical phonons. In a number of high-symmetry crystals, and along the high-symmetry directions, the atomic vibrations are either polarized along the propagation wave vector k, or perpendicular to k. Acoustic modes have one longitudinal acoustic (LA) mode, and two transverse acoustic (TA) modes. A crystal consisting of a unit cell of N-atoms has 3N-3 optical modes. Accordingly, 211 MAX phases have 21 optical modes. In the Fig. 8a, the acoustic branches are shown with red and optical branches are identified with green.
The calculated phonon DOS of Sc 2 SnC is shown in Fig. 8b, revealing that the acoustic and the lower optical modes arise due to the vibration of heavier Sn-atoms. The middle optical branches arise due to the vibration of Sc-atoms. The higher optical branches mostly originate from the vibration of lighter C-atoms. Acoustic phonon is caused by the coherent vibrations of atoms in a lattice outside their equilibrium position. Conversely, the optical phonon originates due to the out-of-phase oscillation of the atom in the lattice when an atom moves to the left and its neighbor to the right. Most of the optical properties of the crystals are controlled by the optical phonons.
Zone-center phonon modes are of particular interest in the lattice dynamics of crystal solids. Since the Snbased 211 MAX phases consist of 8 atoms, they have 24 phonon branches or vibration modes. Three of these are acoustic modes with zero frequency at Γ-point and the remaining 21 are optical modes. Of these 21 optical modes, six are IR active, seven are Raman active and the remaining eight are silent modes. Consistent with the factor group theory, the irreducible representations of the Brillouin zone-center optical phonon modes can be classified as: where A 2u and E 1u are IR active and A 1g , E 1g and E 2g are Raman active and B 1u , B 2g and E 2u are silent modes. The total modes obtained for the M 2 SnC phases in this study are consistent with previous theoretical studies of the lattice dynamics of the various 211 MAX phases [75][76][77] . Each mode has a specific frequency of vibration. Sometimes two or more modes have the same frequency but it cannot be claimed that they are distinct modes; these modes are called degenerate. For this reason, Table 2  Defect processes. The motivation to examine the point defect processes of materials stems from the fact that they can impact the macroscopic materials properties (i.e. radiation tolerance) [78][79][80] . In that respect the investiga- www.nature.com/scientificreports/ tion of point defects in MAX phases is very important as they can be in radiation environments given that they are considered for nuclear applications [81][82][83] . Table 3 lists the defect reactions and the corresponding defect energies for Sc 2 SnC and the existing M 2 SnC MAX phases. In these calculations we have considered all the possible point defects including all the interstitial sites existing in the 211 M 2 SnC MAX phases. The preferable sites are shown in Fig. S1. For the defect reactions we employed the Kröger-Vink notation 84 . In this notation M i stands for an M interstitial defect, V́S n for a Sn vacant site and M Sn an M atom residing in a Sn site (known as antisite defect). Typically, the energies of the Schottky reaction in this system are high (Table 3) and therefore the Frenkel reactions (Table 3, relations 1-3) or the antisite reactions (Table 3, relations 4-6) are more relevant when considering the radiation tolerance of the material. For Sc 2 SnC the C-Frenkel energy is only 3.33 eV inferring that this is not a particularly radiation tolerant MAX phase as compared to most of the other MAX phases considered here (Table 3). Commonly, with the other MAX phases in Table 3 there is the possibility to form antisite vacancies via the recombination of self-interstitials and vacancies. For Sc 2 SnC this is inferred by the negative energies in reactions 7-9 and 12.

Conclusions
In summary, we have employed DFT calculations to investigate the structural, electronic, mechanical and lattice dynamical properties of Sc 2 SnC including defect processes to compare with those of existing M 2 SnC MAX phases. The calculated structural properties show fair agreement with the available experimental values. The structural, mechanical and dynamical stability of Sc 2 SnC is verified. The chemical bonding of Sc 2 SnC is a combination of metallic, covalent and ionic. The softness, elastic anisotropy level and deformability of Sc 2 SnC are moderate compared to the other M 2 SnC phases. Sc 2 SnC has the potential to be etched into 2D MXenes and be a promising Table 2. Theoretical wavenumbers ω and symmetry assignment of the IR-active and Raman-active modes of the Sn-based M 2 SnC MAX phases.

Irr. Rep
Wavenumbers ω (cm 1 )  www.nature.com/scientificreports/ TBC material, similar to the other M 2 SnC phases. The hardness of M 2 SnC, including Sc 2 SnC, follows the trend of bulk modulus rather than shear and Young's modulus while the melting point has a better relationship with the bulk and Young's modulus than with the shear modulus. The rate of declination of lattice thermal conductivity with temperature is almost similar for all M 2 SnC phases. The minimum thermal conductivity shows a linear relationship with the Debye temperature. The highest frequency of the IR active modes for Sc 2 SnC is lowest in the M 2 SnC family while the highest frequency of the Raman active mode is largest for Sc 2 SnC. Examining the defect processes of the existing M 2 SnC phases it is revealed that Sc 2 SnC is less radiation tolerant than numerous 211 MAX phases.

Data availability
All data generated or analysed during this study are included in this published article and its supplementary information files.